Hyperelliptic curves in abelian varieties

Dedicated to Yuri Ivanovich Manin on the occasion of his 60th birthday

Frans Oort  Johan de Jong

Introduction

In this paper we prove the following theorem:

(0.1) Main Theorem: Let g ≥ 3, and let X be a generic abelian variety of dimension g. Then X does not contain a hyperelliptic curve.

(0.2) To be more precise: Consider the moduli space A_{g} of polarized abelian varieties of dimension g. Let K be a field, and let A be an irreducible component of A_{g} ⊗ K. Let η be the generic point of A, and let X be the abelian variety over L := K(A)^{a} corresponding with \overline{η} ∈ A(L) (here superscript a is taking the algebraic closure). Let C ⊂ X be an irreducible algebraic curve. If g ≥ 3, then the normalization C^{∼} is not a hyperelliptic curve.

This theorem in characteristic zero was proved by Pirola, see [18], Section 2, Theorem 2; see [19], page 238. The proof by Pirola follows from the fact that hyperelliptic curves in an abelian variety in characteristic zero are “rigid up to translation”. In positive characteristic this rigidity is no longer true, and we analyze the situation there. For surfaces, see [20], Section 2. These results by Pirola and by Schoen inspired us to give the proofs below. In [17] we analyzed the proof by Pirola, and some details of that paper will be used and recorded here.

1 Notation

(1.1) Definition: We say that C is hyperelliptic or quasi-hyperelliptic (together abbreviated as (q)hyperelliptic) if this is a geometrically integral curve over a field, such that its normalization C^{∼} has genus zero, or genus one, or it admits a separable morphism of degree two onto a rational curve.

(1.2) Lemma: Let K ⊂ L be an extension of fields, and C a curve over K; this curve is hyperelliptic if and only if C ⊗ L is hyperelliptic. Proof: For a curve of genus at least two, use the fact that the canonical morphism is a double covering onto a rational curve if and only if the curve is hyperelliptic; this property is true over K if and only if it is true over L.

(1.3) Lemma (see [20], Lemma 1.1): Let C be a (q)hyperelliptic curve, and g : C → C′ a dominant morphism. Then C′ is (q)hyperelliptic. Proof: If C′ is rational or elliptic the conclusion is clear. We can work over a field extension. Hence we can suppose that a Weierstrass point W ∈ C(K) rational over K exists. We can use the point W to define the Albanese mapping

f : C ↪ J = Jac(C), W ↦ 0.

Then multiplication by −1 on J maps f(C) into itself, and it coincides with the hyperelliptic involution on C. Use g(W) to construct the Albanese map for C′. The Albenese property implies that the −1_{J} ∈ End(Jac(C′)) maps the image of C′ to itself, hence C′ is hyperelliptic.

2 Rigidity in positive characteristic.

We study families of hyperelliptic curves in an abelian variety. Such families should be considered up to translation, we do this by rigidifying such maps by mapping a marked point to 0 ∈ X. For hyperelliptic curves we take such a marked point equal to a Weierstrass point. We shall see that non-trivial families of hyperelliptic curves exist only inside a supersingular abelian variety.

(2.1) Notation: Let K be a field. Let T be an irreducible scheme over K, let f : C → T be a proper and smooth relative curve, and let w : T → C be a section of f. Let X be an abelian variety over K, and let

φ : C → X × T

be a T -morphism such that

(φ·w : T → X × T) = 0 × 1_{T}

is the zero morphism. We write

ψ = (C ⟶^{φ} X × T ⟶^{proj_{1}} X).

If every fibre of f : C → T has genus zero, the morphism ψ is constant i.e. ψ(C) = 0, see [2], p. 107, Corollary 3.8. If every fibre of f : C → T has genus one, we obtain a family of abelian subvarieties of X, which is constant, see [8], page 26, Theorem 5, or use rigidity as in [2], p.146, Proposition 20.3; in this case the morphism ψ is either constant, or its image ψ(C) = E is an elliptic curve contained in X. We are interested in cases where the image ψ(C) has higher dimension, i.e. the case of non-trivial families of curves in X. From now on we suppose that the genus of C/T is at least two.

Suppose ι ∈ Aut(C/T) is an involution so that every fibre of C/ < ι >→ T is a rational curve, i.e. every fibre of f is hyperelliptic. Suppose w : T → C is a section of f such that w(T) is contained in the fixed point locus of ι, i.e. the points of w(T) consist of Weierstrass points of the fibres.

The condition that

(φ·w : T → X × T) = 0 × 1_{T}

is the zero morphism (i.e. every marked Weierstrass point is mapped onto 0 ∈ X) implies that the diagram

C ⟶ X
ι↓  ↓ −1_{X}
C ⟶ X

is commutative.

(2.2) Theorem: Suppose char(K) = p > 0. Suppose T is an absolutely irreducible curve over K. We use the conditions described above. Suppose that the image of

C ⟶^{φ} X × T ⟶^{proj_{1}} X

is a surface, i.e. we have a non-trivial family of hyperelliptic curves in X. Let Y ⊂ X be the smallest subabelian variety containing ψ(C). Then Y is supersingular.

(2.3) Proof: We proceed in various steps. Using the fact that family C → T maps onto a surface in X we show (5): the image H^{2}(X) → H^{2}(S) consists of Tate classes. From there the result follows. First Step (1) It suffices to show the result in case all data are defind over a finite field. We may assume that the data T, C, w, X, φ, ψ and Y are defined over a scheme U of finite type over F_{p}. If for every closed point u ∈ U the abelian variety Y_{u} over κ(u) is supersingular, then Y defines a supersingular abelian variety over the function field of U, and we are done. In this way we reduce to the case where k is a finite field.

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In the proof we shall take several times a finite extension of this finite field (and denote the new base field again by k) in order to get rid of roots of unity that occur in eigenvalues of Frobenius. The symbol Q_{ℓ}(−1) denotes a one dimensional vector space on which Frobenius, x ↦ x^{q}, acts by q = #(k). An element η ∈ H^{2}(S) is called a Tate class if it is an eigenvector with eigenvalue q·(some root of 1); then, by enlarging k we achieve that this eigenvalue equals q (= the number of elements of the new field).

Second Step. Notation: we write

S := (ψ(C))^{Zar.Closure} ⊂ X.

Note that there exists an involution ι ∈ Aut(C/T) and that ψ is equivariant between ι ∈ Aut(C/T) and −1 ∈ Aut(X).

(2) There exists a complete, regular surface S̃, an involution ι̃ ∈ Aut(S̃), a birational dominant map S̃ → C and an equivariant morphism ψ̃ : S̃ → S ⊂ X extending ψ : C → S. We replace T by its normalization in a finite extension of the function field of T. Thus, by the stable reduction theorem, use [4], Corollary (2.7) , we may assume that C extends to a stable curve C̄ over a projective smooth completion T̄ of T. The section w extends to w̄ over T̄; the involution ι extends to ῑ over T̄, see [4], Lemma (1.12). Let S̃ be a blow up of the surface C̄ such the rational map ψ extends to a morphism ψ̃ : S̃ → S and such that ῑ lifts to ι̃ on S̃. We can choose S̃ to be regular.

□

We fix a prime number ℓ different from p. If W is a scheme of finite type over k, we put H^{i}(W) := H_{ét}^{i}(W ⊗_{k} k̄, Q_{ℓ}).

Note that the scheme C̄/⟨ῑ⟩ exists and is a semi-stable curve of genus 0 over T̄. Note (after extending the base field) that H^{2}(C̄/⟨ῑ⟩) = Q_{ℓ}(−1)^{r} for some r ≥ 0, by considering the Leray spectral sequence for C̄ → T̄.

Third Step (3) The morphism S̃ → S induces H^{2}(S) → H^{2}(S̃). The kernel of this homomorphism consists of Tate classes. This is a general fact. Take the Stein factorization

S̃ →^{f} S′ → S

of ψ. Since the morphism S′ → S is finite and dominant the map H^{2}(S) → H^{2}(S′) is injective on elements of weight two (analogous argument as below). We write f̄ = f × k̄. We see that f : S̃ → S′ is a resolution of singularities of a normal surface S′. In this case we have (with obvious notations) that R^{0}f̄_{∗}Q_{ℓ} = Q_{ℓ}, and R^{1}f̄_{∗}Q_{ℓ} = ⨁(i_{p})_{∗}H^{1}(f^{−1}(p)), and R^{2}f̄_{∗}Q_{ℓ} = ⨁(i_{p})_{∗}H^{2}(f^{−1}(p)). The Leray spectral sequence associated to f̄ gives an exact sequence

H^{0}(S′ ⊗_{k} k̄, R^{1}f̄_{∗}Q_{ℓ}) ⟶ H^{2}(S′ ⊗_{k} k̄, R^{0}f̄_{∗}Q_{ℓ}) ⟶ H^{2}(S̃)
||  ||  ||
⨁ H^{1}(f^{−1}(p))  ⟶  H^{2}(S′)  ⟶ H^{2}(S̃).

Thus the kernel of H^{2}(S′) → H^{2}(S̃) is the image of the map ⊕H^{1}(f^{−1}(p)) → H^{2}(S). Any element of weight 2 in the H^{1} of a projective curve is a Tate class. □

Fourth Step (4) The image of the homomorphism H^{2}(X) → H^{2}(S̃) consists of Tate classes. [Here we use the property that the image of C in X is a surface, i.e. the condition that the family of hyperelliptic curves inside X is “not constant up to translation”.] Note that H^{2}(S̃) = H^{2}(C̄) ⊕ Q_{ℓ}(−1)^{s} for some s ≥ 0, in view of the fact that the surface C̄ has only rational singularities. Note that the action of −1_{X} on H^{2}(X) is trivial, hence the map H^{2}(X) → H^{2}(S̃) lands in the invariant part of H^{2}(S̃). We combine this with the fact that H^{2}(C̄)^{⟨ῑ⟩} = H^{2}(C̄/⟨ῑ⟩) = Q_{ℓ}(−1)^{r}, to conclude that the image of H^{2}(X) → H^{2}(S̃) consists of Tate classes.

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Fifth Step (5) The image of the homomorphism H^{2}(X) → H^{2}(S) consists of Tate classes. This follows from (3) and (4).

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Sixth Step (6), Lemma: Let λ_{1}, ⋯ λ_{4} ∈ C be q-Weil numbers such that for every 1 ≤ i < j ≤ 4 we have λ_{i}·λ_{j} = q·(some root of 1). Then for every 1 ≤ i ≤ 4 we have λ_{i}^{2} = q·(some root of 1). Let i = 1 (renumber if necessary), and consider

λ_{1}^{2}·(λ_{1}·λ_{2})(λ_{3}·λ_{4}) = (λ_{1}·λ_{2})(λ_{1}·λ_{3})(λ_{1}·λ_{4}).
□

Seventh Step (7) We choose an isogeny X → X′ × X′′ such that X′ is supersingular, and such that X′′ does not contain a supersingular abelian subvariety. Note that this implies: for any eigenvalue λ of Frobenius on X′′ and for any n ∈ Z_{>0} we have λ^{2n} ≠ q^{n}. We are going to show that the image of S ⊂ X in X′′ is the zero point; in that case the smallest abelian subvariety S ⊂ Y ⊂ X maps to an abelian subvariety of X′, i.e. Y is supersingular.

Let η_{S} ∈ H_{4}(X) = (Λ^{4}H^{1}(X))^{∧} denote the fundamental class of S in X. Consider the pairing

⟨ , ⟩_{S} : H^{2}(X) × H^{2}(X) ⟶ Q_{ℓ}

given by

⟨a, b⟩_{S} := η_{S}(a ∧ b) = ∫_{S} a|_{S} ∧ b|_{S}.

By the above this pairing has the following property:

(P) : if a ∈ H^{2}(X) is an eigenvector of Frobenius and a is not Tate (i.e., the eigenvalue is not equal to q), then the linear form ⟨a, −⟩_{S} on H^{2}(X) is zero.

Let e_{1}, …, e_{2g} be a basis of H^{1}(X) consisting of eigenvectors of Frobenius with eigenvalues λ_{1}, …, λ_{2g}. The dual basis is written e_{1}^{∗}, …, e_{2g}^{∗}; these have the eigenvalues λ_{1}^{−1}, …, λ_{2g}^{−1}. We can write

η_{S} = ∑_{1≤i_{1}<i_{2}<i_{2}<i_{4}≤2g} α(i_{1}, …, i_{4})e_{i_{1}}^{∗} ∧ e_{i_{2}}^{∗} ∧ e_{i_{3}}^{∗} ∧ e_{i_{4}}^{∗}.

Since η_{S} is a Tate class in H_{4}(X), we see that the coefficient α(i_{1}, …, i_{4}) is zero unless λ_{i_{1}}λ_{i_{2}}λ_{i_{3}}λ_{i_{4}} = q^{2}. The condition (P) implies that

α(i_{1}, …, i_{4}) ≠ 0 ⇒ λ_{i_{s}}λ_{i_{r}} = q,

for all s, t ∈ {1, 2, 3, 4}, s ≠ t. By the previous lemma we see that this implies that only the basis elements e_{i}^{∗} with λ_{i}^{2} = q are used in a minimal epression of η_{S} as above.

We choose a very ample effective divisor H′′ ⊂ X′′; denote the inverse image by H ⊂ X. We see that the element η_{H} can be written using only the duals of the basis elements e_{i} whose eigenvalue λ_{i} does not satisfy λ_{i}^{2} = q, and hence the element η_{S} ∧η_{H} = 0 of H_{2}(X) is zero. This implies that the class in H_{2}(X) of the intersection S ∩H is zero. We conclude that (a translate of) H does not meet S, hence the image of S in X′ × X′′ is contained in X′ = X′ × {0}. Thus Y ⊂ X as in the theorem maps to an abelian subvariety of X′. This concludes the proof of Theorem (2.2).

(2.4) The analogue of the theorem above in case for the base field C (well known by work of Xiao, see [21], and see Pirola, [18], [19]) can be paraphrased using Hodge theory to deal with the case that the field k = C; the end result will be that Y = (0), hence that such a family does not exist. (In this case one should argue with Hodge-types of basis elements instead of Frobenius eigenvalues.)

(2.5) The analogue of the theorem above in case for the base field C can also be proved using the characteristic p result. Indeed, suppose T, C, w, X, φ, ψ and Y are as in the theorem and are defined over C. By a specialization argument as before, we see that we may assume they are defined over a number field. By reducing modulo finite primes and invoking (2.2) we see that Y is an abelian variety which is supersingular at almost all primes. However by [3], page 372, Corollary (2.8) such abelian varieties do not exist, and we have shown that a hyperelliptic curve in an abelian variety in characteristic zero is rigid (up to translation).

(2.6) Remarks: 1) In Manin, [11], page 35, “Classification theorem for formal groups up to isogeny” we find a way to describe an isogeny invariant for abelian varieties in positive characteristic (over an algebraically closed field). Iso-simple p-divisible groups are classified by two integers, which are relatively prime, the dimension and the dimension of the Serre-dual.

2) An elliptic curve is called supersingular if it is defined over a field of positive characteristic p, and if it has no geometric point of order exactly p. An abelian variety is called supersingular if it is isogenous over an algebraicaly closed field with a product of supersingular elliptic curves. We write G_{1,1} for the formal group of a supersingular elliptic curve over an algebraically closed field (see [11], page 35).

3) Suppose that in the formal isogeny type of an abelian variety Y only the factors G_{1,1} appear. Then there is an isogeny Y ∼ E^{g}, where E is a supersingular elliptic curve, see [15], Theorem 4.2. Note that in general splitting of the formal group of an abelian variety does not imply splitting of the abelian variety (up to isogeny), in fact any symmetric formal isogeny type which is not supersingular can be realized by a simple abelian variety, see [9].

We consider the question whether a situation as in the theorem can be realized.

4) By [5], Corollary 4.2 we see that for p = 2, for any n ∈ Z_{≥1} and g := 2^{n} there exists a family of dimension n of hyperelliptic supersingular curves of genus g. Such a family is given by

Y^{2} + Y = X · ∑_{i=0}^{i=n+1} a_{i}X^{2^{i}}.

If 2 ≤ g ≤ 4 the situation as in the previous theorem can be realized.

5) Indeed, for g = 2 by [12], Exp. VIII there exists a non-isotrivial family C → P^{1} of supersingular curves of genus two. There is an isogeny Jac(C) → (E^{2}) × P^{1}, where E is a supersingular elliptic curve, and we obtain a nontrivial family of (hyperelliptic) curves of genus 2 in Y = X := E^{2}.

6) Along analogous lines we find for g = 3 a subset S_{3} ⊂ M_{3} ⊗ F_{p} of supersingular curves of genus 3. This intersects the hyperelliptic locus in components of dimension 1, see [16], and we obtain nontrivial families of hyperelliptic curves of genus 3 in E^{3} in all characteristics.

7) For g = 4 we find S_{4} ⊂ A_{4,1} ⊗ F_{p}, with dim(S_{4}) = [g^{2}/4] = 4, see [10], 4.9i. The hyperelliptic locus for g = 4 has dimension 2g − 1 = 7, either taken as a subset of M_{4} ⊗ F_{p} or as the closure B_{4} of the image of the hyperelliptic locus in A_{4,1} ⊗ F_{p}. Suppose we can find at least one hyperelliptic (irreducible), supersingular curve of genus 4, with Jacobian (J, λ). The intersection S_{4} ∩ B_{4} contains (J, λ) and has components of dimension at least 7 + 4 - (4(4+1)/2) = 1. Hence we see that in such a case we can construct a nontrivial family of hyperelliptic curves of genus 4 in E^{4}.

8) Here is an example of a hyperelliptic, supersingular curve of genus 4: choose a prime number p > 2 with p mod 5 ∈ {2, 3, 4}, and consider the hyperelliptic curve C of genus 4 defined over F_{p} by η^{2} = ξ^{10} − 1. Note that this curve has two involutions: ξ ↦ −ξ, and η ↦ ±η. Both define a double covering C → D over a finite extension of F_{p}, where D is defined by Y^{2} = X^{5} − 1. One shows that these maps induce an isogeny

Jac(C) → Jac(D) × Jac(D).

By [7], Prop. 1.13 we know that D is hyperelliptic; hence we conclude that C is a hyperelliptic, supersingular curve of genus 4.

In fact our construction is a special case of the following: choose mutually different points λ_{1}, ⋯, λ_{5} ∈ P^{1} − {0, ∞}, construct f : D → P^{1}, a double cover ramified in {λ_{1}, ⋯, λ_{5}, ∞}, use that the divisor class f^{−1}(0) ∼ 2.W, where W is the Weierstrass point on D lying over ∞ ∈ P^{1}, and construct C → D, a double covering ramified in f^{−1}(0). If {λ_{1}, ⋯, λ_{5}, ∞} and {0, λ_{1}, ⋯, λ_{5}} both define a supersingular curve of g = 2, then C is a hyperellliptic, supersinglar curve of genus 4.

(2.7) For g ≥ 5 we have 2g − 1 + [g^{2}/4] − (g(g + 1)/2) ≤ 0. This limits the methods in the previous remark to g ≤ 4. Indeed: Question: For which g ≥ 5 and for which p does there exist a supersingular abelian variety Y of dimension g in characteristic p and a nontrivial family of hyperelliptic curves in Y spanning Y? Question: For which g and for which p does the intersection S_{g} ∩ B_{g} contain a component of positive dimension meeting the image of M_{g} → T_{g}^{0} ⊂ A_{g,1}? [Notation: S_{g} ⊂ A_{g} ⊗ F_{p} is the supersingular locus, and B_{g} ⊂ A_{g} is the closure of the set of Jacobians of hyperelliptic curves.] We do not know whether such examples do exist except for the cases recorded above.

3 Proof of the Main Theorem.

(3.1) Suppose the Main Theorem is true in positive characteristic, then it follows in all characteristics. Indeed, we consider A_{g} → Spec(Z). By [14], Corollary 3.2 we know that a generic abelian variety X in characteristic zero can be obtained as the lifting of a generic abelian variety X_{0} in positive characteristic. Suppose X would contain as hyperelliptic curve C ⊂ X; note that X is simple, hence C generates X. The schematic closure C_{0} ⊂ X_{0} would contain an irreducible component which is a hyperelliptic curve, a contradiction. This proves the claim.

(3.2) Proof of the Main Theorem in positive characteristic, see [18], Section 2, see [17], Section 6. In this part of the paper we consider fields of characteristic p, where p is a fixed prime number. We fix g ≥ 3, we consider a generic abelian variety X as in (0.2), note that X can be defined over a finite extension of K(A). Suppose there would exist a hyperelliptic curve C ⊂ X contained in X (and we are going to obtain a contradiction). Note that X is absolutely simple, hence C generates X. Suppose that C ⊂ X is chosen in such a way that there is no hyperelliptic curve of smaller genus in X. This curve is defined over a finite extension of K(A). Let D = C^{∼} denote the normalization of C. We write h for the genus of D. Over some finite extension of this field at least one of the Weierstrass points of D, call it W, is rational. We map D → Alb(D) =: Y in such a way that W maps to zero, and we translate C in X so that the image of W in C ⊂ X is equal to 0 ∈ X. We obtain a homomorphism f′ : Alb(D) = Y → X, which is non-constant, hence surjective (X being simple). Hence there exists an abelian variety Z and a homomorphism f′′ : Y → Z such that f = (f′, f′′) : Y → X × Z is an isogeny (complete reducibility by Poincaré and Weil). The curve D has good reduction over an open set. Putting together all information just discussed we arrive at: First Step (1): There exists a non-empty Zariski-open subset U′ ⊂ A, a finite surjective morphism U → U′, we denote the universal family by X → U, i.e. X\overline{η} = X, we have a smooth curve D → U with geometric generic fibre D, a section W : U → D giving a Weierstrass point in every fibre, abelian schemes Y and Z over U extending Y and Z, and morphisms

D ↪ Jac(D) = Y ⟶^{f} X ×_{U} Z ⟶^{proj_{1}} X .

We write n := deg(f).

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Second Step: We choose an abelian variety B of dimension g − 1 which is ordinary. Let Λ = Λ(B) be the set of all points in U corresponding over some algebraically closed field with a polarized abelian variety (X′, λ′) such that there exists some elliptic curve E and an isogeny X′ ∼ B × E with degree prime to p. The set Λ(B) ⊂ U is Zariski-dense, by a result by Chai, see [1], Theorem 2 on page 477. Note that Λ is a countable union of curves,

Λ = ⋃_{α∈A} Λ_{α}.

Consider

D |_{Λ_{α}} =: D_{α} → X |_{Λ_{α}}→ B × E

(this is defined over a finite cover of Λ_{α}, here E is a non-constant family of elliptic curves). Claim (2): The family D |_{Λ_{α}} is non-constant. In fact, suppose it is constant, projecting it onto E we obtain a surjective morphism to a nonconstant family of elliptic curves, the dual of Alb(D |_{Λ_{α}}) → E produces a morphism of abelian schemes, by rigidity the images of elliptic curves are constant, a contradiction. Consider

D_{α} → X |_{Λ_{α}}→ B × E → B × Λ_{α}.

Because B is ordinary we conclude from (2.2), and from the hypothesis that D is hyperelliptic that the image of D_{α} → B × Λ_{α} is constant; we denote the image by G_{α}.

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Third Step , Claim (3): There is an integer N such that for every α and for every t ∈ Λ_{α} the degree is bounded:

deg(D_{t} → G_{α}) ≤ N.

In fact, let i be choosen such that p^{i−1} does but p^{i} does not divide the degree n = deg(f), where f : Y → X × Z is as in the first step. For every t as above we have a factorization

D_{t} → I → G_{α},

where the first map is purely inseparable, and the second is separable.

First we remark that the degree of the first, purely inseparable map is less than p^{i}. Indeed, if H = D_{t}, and F^{i} : H → H^{(p^{i})} would factor H → B, then for Y_{t} = Alb(D_{t}) =: J → J^{(p^{i})} would factor the morphism

J → X_{t} × Z_{t} → X_{t} → B × E_{t} → B,

a contradiction with the fact that p^{i} does not divide n, and that X_{t} → B × E_{t} is separable. Hence this bounds the inseparable degree.

Fourth Step , Claim (4): Next we note that the genus h_{α} of G_{α} is stricly less than h. In fact, D_{α} → G_{α}×Λ_{α} maps a non-constant curve of relative genus h to a constant curve of genus h_{α}. If h would be equal to h_{α}, the generically separable morphism D_{α}^{(p^{s})} → G_{α} × Λ_{α} with some s ≤ i − 1 would give a generically separable surjective map from a non-constant abelian scheme to constant abelian variety of the same relative dimension, hence an isogeny, dually this would give an isogeny from a constant to a non-constant one, with kernel an extension of an étale group scheme by the dual of an étale group scheme; such a finite group scheme is rigid in this constant group scheme, and we have a contradiction. Hence

genus(G_{α}) =: h_{α} < h := genus(D).

Note moreover that G_{α} generates B, hence

2 ≤ g − 1 ≤ h_{α}.

Putting things together we see that the separable degree

deg(I → G_{α}) ≤ (2h − 2)/(2g − 4) ≤ h − 1.

This finishes the proof that for every α and for every t the degree deg(D_{t} → G_{α}) is bounded. Hence the proofs of Step 3 and Step 4 are concluded.

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Fifth Step (5). Under the assumption made in the first step we are going to construct a hyperelliptic curve F ⊂ X of genus smaller than h in X. In this part we use a variant of the “Mori trick”, see [13], pp. 600/601.

Using the results of the previous step we see we can choose a subset A′ ⊂ A, an integer n ≤ N (see Step 3), and an integer i with g − 1 ≤ i < h such that

Λ′ = ⋃_{α∈A′} Λ_{α}

is dense in U, such that for every α ∈ A′ we have genus(G_{α}) = i and such that

deg(D_{t} → G_{α}) = n.

Let F → M be a fine moduli space of curves of genus equal to i in characteristic p. Consider the relative curve D × M → U × M (which is projective and flat) and the relative curve U × F → U × M (which is projective, say by considering a multicanonical morphism on F). By Grothendieck, see [6], 4.c, pp. 221-19/20 we conclude that the functor

HOM_{U×M}(D × M, U × F)

is representable by a disjoint union of schemes of finite type indexed by Hilbert polynomials. Let P be the part given by morphisms which are of degree n on the fibres. For every α ∈ A′ the inverse image of Λ_{α} in P → U × M → U is non-empty. Hence the morphism P → U has non-empty fibres over a dense set in U. Hence there exists a component of P which is dominant over U. Hence over the geometric generic point \overline{η} there exists a covering D → F which, going over to a finite cover of U, restricting to a (possibly smaller dense open) subset (again denoted by U), give a smooth family F → U, and a covering D → F which specializes to each of the morphisms D_{α} → F_{α}, with α ∈ A′ (for those Λ_{α} meeting the new U) considered above.

Consider

φ : Jac(D) → X

and

ψ : Jac(D) → Jac(F)

defined over K(U). By taking a finite extension of this field we have that T := ((Ker(ψ))^{o})_{red} is an abelian variety. By the Poincaré-Weil complete reducibility theorem we conclude that there exists a positive integer s, and a morphism g : Jac(F) → X such that ψ·g = s_{Jac(F)}. Hence we obtain

ρ : Jac(F) → X, such that ρ·ψ = (s·1_{X})·φ.

There are two cases:

(1) either ρ ≠ 0, or (2) ρ = 0.

In case (1), note that D → F is surjective, hence F is hyperelliptic (see 1.3). Note that i = genus(F) < genus(D) = h. Hence (1) gives a contradiction with the minimality assumption of the degree of a map from a (q)hyperelliptic curve to X made in step one.

Assume case (2), ρ = 0. By rigidity of abelian varieties we see that for any t ∈ U the map ρ_{t} : Jac(F_{t}) → X_{t} is zero. However for t ∈ Λ_{α} as above, the map Jac(F_{t}) → B is not zero (it is even surjective). This shows that ρ_{t} ≠ 0. Using rigidity of abelian variety we would obtain ρ ≠ 0, a contradiction. This finishes the proof of the Main Theorem in positive characteristic, hence by (3.1) we conclude the Main Theorem in all characteristics.

References

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Frans Oort Johan de Jong Mathematisch Instituut Department of Mathematics Budapestlaan 6 Science Center, 1 Oxford Street NL - 3508 TA Utrecht Cambridge MA 02138 The Netherlands U. S. A. email: oort@math.ruu.nl email: dejong@abel.math.harvard.edu